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Why do you have to include the Jacobian for every coordinate system, but the Cartesian?

In physics and engineering it is common to convert between different coordinate systems - spherical, polar, Cartesian, e.t.c. - depending on the problem. Physically, they are all clearly equivalent and it shouldn't matter which one we use.

Nevertheless, when solving problems involving volume or surface integrals, we always have to add the Jacobian matrix to the expression if we are any other coordinate system but Cartesian, just as if we were converting from the Cartesian system.

Take the expression: $$ \iiint_V \nabla \cdot \textbf{u}\ dV $$

It seems intuitive that we should be able to go from this expression and express $V$, $\textbf{u}$, $\nabla$, and $dV$ in whatever coordinate system we want at this point, but plugging the spherical coordinates straight into the expression yields the wrong answer - one has to add the Jacobian just as if the first step was expressing everything in Cartesians and converting to polars. Why do Cartesians get such special treatment? My expectation would be because the eigenvectors are constant throughout space, but I would appreciate a more thorough explanation.